Class 4b: Simple Linear Regression diagnostics

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.title[
# Class 4b: Simple Linear Regression diagnostics
]
.author[
### Business Forecasting
]

---

</style>

---
## Roadmap

### This class
- Testing assumptions behind residuals
  - Linearity
  - Constant Variance
  - Uncorrelated Residuals
  - Normality
  
---

Let's revisit the assumptions behind linear model:

0. Model is linear in the parameter and with additive error term
1. `$E(u_i)=0$`
2. `$E(u_i|x)=0$`
3. `$Var(u_i)=\sigma^2$`
4. `$cov(u_i,u_j)=0$`

Additional assumption needed for hypothesis testing and confidence intervals:

5. `$u_i \sim N(0,\sigma)$`

---
### Assumptions

Visualizing residuals, we can test: 
- Linearity
- Constant Variance (homoskedasticity)
- Uncorrelated errors

---

With vizuale and numerical tests we can analyze:
- Normality

---

### Linearity diagnostic

1. Plot residuals against fitted values of y `$\hat{y}$`
2. Check if residuals have systematic non-linear pattern

---
### Linearity diagnostic

---
### Constant variance (homoskedasticity)  diagnostic

1. Plot residuals against fitted values of y `$\hat{y}$`
2. Check if variance changes as `$\hat{y}$` changes (heteroskedasticity)

---
### Constant variance (homoskedasticity) diagnostic

---
What goes wrong here?

<img src="data:image/png;base64,#C_4_slides_a_files/figure-html/unnamed-chunk-5-1.png" width="100%" />
---

What goes wrong here?

---
### Correlation of error terms

1. Plot fitted residuals vs their value in previous observation `$e_i$` vs `$e_{i-1}$`
2. So `$e_2$` vs `$e_1$`, `$e_5$` vs `$e_4$` etc

---
### Correlation of error terms

1. We can also calculate the correlation with other lags
  - Example: `$e_i$` vs `$e_{i-3}$` 
  - Note down correlation at each lag: `$\small \rho(e_i,e_{i-1})$`, `$\small  \rho(e_i,e_{i-2})$`, `$\small  \rho(e_i,e_{i-3})$`

<img src="data:image/png;base64,#C_4_slides_a_files/figure-html/unnamed-chunk-8-1.png" width="100%" />
---

### Normality

We can start by looking at histograms
- Some are obvious

---
### Normality
We can start by looking at histograms
- Some are obvious

---
### Normality
We can start by looking at histograms
- Some are **NOT** obvious
- Then we need a different test

---
### Normality

**Q-Q Plot**: Quantlie-Quantile Plot

- Comparing quantiles of our data to what they should be if they come from normal distribution

- Procedure:
  - Standardize residuals

--
  - Sort them

--
  - Check which quantile they represent
    - This is equivalent to checking cumulative probability: `$CP_i=\frac{Index}{n+1}$`
    - Where index is the number of observation if we sort if from smallest to  largest 
        - 1 is the smallest, 2 is the second smallest, ..., n is the largest
    - In other words, what share of data is smaller than this observation

--
  - Compare them to quantiles from standard normal

--
  - If our distribution is normal, the quantiles should be similar

---
### Q-Q plot

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---
### Q-Q plot

- Next, we plot the sample quantiles vs the standard normal quantiles
- If they are similar, they should be on the straight line

### Q-Q plot
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---
### Q-Q plot
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---
### Q-Q plot

---
### Jarque-Bera Test

- We can also use the **Jarque-Bera Test** to see whether sample comes from a normal distribution

**Intuition**
- Skeweness of normal distribution is 0 (it's symmetric)

`$$S = \frac{\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^3}{\left(\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^2\right)^{\frac{3}{2}}}$$`

---

---
### Jarque-Bera Test

- We can also use the **Jarque-Bera Test** to see whether sample comes from a normal distribution

**Intuition**
- Skeweness of normal distribution is 0 (it's symmetric)

`$$S = \frac{\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^3}{\left(\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^2\right)^{\frac{3}{2}}}$$`

- Kurtosis of normal distribution is 3 
- Excess kurtosis is 0

`$$EK = \frac{\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^4}{\left(\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^2\right)^{2}} - 3$$`

---

---
### Jarque-Bera Test

- We can also use the **Jarque-Bera Test** to see whether sample comes from a normal distribution

**Intuition**
- Skeweness of normal distribution is 0 (it's symmetric)

`$$S = \frac{\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^3}{\left(\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^2\right)^{\frac{3}{2}}}$$`

- Kurtosis of normal distribution is 3 
- Excess kurtosis is 0

`$$EK = \frac{\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^4}{\left(\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^2\right)^{2}} - 3$$`

- We will test, whether our sample has (more or less)
  - Skewenss of 0
  - Kurtosis of 3

---
### Jarque-Bera Test

The test statistic for our test is:

`$$JB = \frac{n}{6} \left(\frac{S^2}{2} + \frac{EK^2}{4}\right)$$`

- Its value will be high if:
  - Skeweness deviates significantly from 0
  - Kurtosis deviates significantly from 3

- If the data really comes from normal (that's our null hypothesis), then:

`$$JB \sim  \chi_2$$`
- It follows the Chi-squared distribution with 2 degrees of freedom.

---
### Jarque-Bera Test

So in our usual testing setting:

- `$H_0: e_i$` comes from normal (JB is small)
- `$H_A: e_i$` Does not come from normal (JB is large)

- It's a one sided test, so we reject at `$\alpha$` if

`$$JB_{test}>\chi_{1-\alpha,2}$$`
---
### Jarque-Bera Test

```r
library(tseries)
jarque.bera.test(sample_data)
```

```
## 
## 	Jarque Bera Test
## 
## data:  sample_data
## X-squared = 0.96619, df = 2, p-value = 0.6169
```

---
### Jarque-Bera Test

```r
library(tseries)
jarque.bera.test(sample_data)
```

```
## 
## 	Jarque Bera Test
## 
## data: sample_data
## X-squared = 420.19, df = 2, p-value < 2.2e-16
```

---